Entrapment efficiencies of hydrodynamic boundary layers on rising bubbles

Flotation and separation practice shows that fine hydrophilic solids are often drawn into the froth product. The occurrence of this unwanted event in the classical froth flotation has led to the idea of using it for the separation by size of ground materials. Thus, a method for the extraction of hydrophilic fines by foaming of a suspension was proposed. The aim of the present study is to relate this phenomenon to the residence time of the particles in the vicinities of the rising bubbles. Dynamic interactions of fine solids with rising bubbles cause perturbations in the background flow field. A procedure for the mathematical modeling of these disturbance effects is proposed. The initial idea is that the particles lag behind the background bubble-driven flows. A key point is the possibility of classifying fine entities according to a general criterion, containing only parameters of the outer flow. The basic result is that there exists a range of particle and bubble dimensions for which this entrapment is optimal. The proposed model investigation gives a concise explanation for the observed capture of fine solids in many flotation and separation processes.     

Conformally invariant tensors of an almost Hermitian manifold associated with the holomorphic curvature tensor

We consider an almost Hermitian manifold and apply the conformal change of metric to its holomorphic curvature tensor. In such a way we find that the generalized Bochner curvature tensor can be expressed as a linear combination of B ₁, B ₂, and B ₃ such that (6.4) holds. Each of the tensors B ₁, B ₂, B ₃ is conformally invariant and satisfies the condition (1.2) of K?hler type.     

Adsorption layer properties and foam film drainage of aqueous solutions of tetraethyleneglycol monododecyl ether

The aim of the present study is to clarify how the surfactant adsorption layer properties are related to the course of the drainage parameters of microscopic foam films in the special case of aqueous solutions of the non-ionic amphiphile tetraethyleneglycol monododecyl ether (C₁₂E₄), containing premicellar nanostructures. The scope of the research covers adsorption dynamics, construction of equilibrium adsorption isotherms, studies on surface rheology of the interfacial layers and microscopic foam film drainage kinetics. It is established that in the premicellar concentration domain considerable irregularities of the adsorption layer properties are observed: two plateau regions are registered in the experimental surface tension isotherm along with unusual changes of the surface rheological characteristics. The systematic investigation of the drainage of microscopic foam films obtained from these solutions show that the dependencies of basic kinetic parameters of the films on the amphiphile concentration run in synchrony with the changes in the adsorption layer properties. This fact is related to the presence of smaller surfactant aggregates (premicelles). They are presumed to be organized as Platonic bodies. The premicelles play also a significant role in the kinetic stability of the films. The importance of this research is in providing better insight into the initial stages of self-assembling phenomena and into the factors determining the adsorption layer properties and the drainage behaviour of thin liquid films.     

Problem of Measurement Within the Operator Formulation of Hybrid Systems

The basic concepts of classical mechanics are given in operator form. Then ahybrid systems approach with the operator formulation of both quantum andclassical sectors is applied to the case of an ideal nonselective measurement. Itis found that the dynamical equation, consisting of the Schrödinger and Liouvilledynamics, produces noncausal evolution when the initial state of the measured(quantum mechanical) system and measuring apparatus=nclassical mechanicalsystem is chosen to be as demanded in discussions regarding the problem ofmeasurement. Nonuniqueness of possible realizations of the transition from apure noncorrelated to a mixed correlated state is analyzed in detail. It is concludedthat the state of the quantum mechanical system instantaneously collapses becauseof the nonnegativity of probabilities, and a dynamical model of this reductionis proposed.     

Effective zones of hydrodynamical interaction of fluid particles at small separations

A system of two approaching fluid particles at small gap widths is investigated. The concept of an effective zone of interaction is used to analyse the stationary Navier-Stokes equations in the respective phases. Particular cases are distinguished in which it is possible to make significant simplifications in the governing differential equations. A validity criterion for the lubrication theory equations in the thin layer is proposed, which accounts for the tangential mobility of the interfaces.     

Hydrodynamical interaction of two emulsion droplets at small separations

The fluid's motion inside emulsion droplets is analysed when they mutually approach along their common axis and a thin liquid film is formed outside. A qualitative flow pattern is presented. Two particular cases are treated — a creeping motion and a boundary layer flow inside the droplets. Estimates are made for the tangential velocity at the droplet/film interface, for the drag force and for the energy dissipated in the respective phases.     

Potential theory in the problems of interfraction at small gap widths. A solid sphere and a droplet

A solid sphere approaches a spherical droplet at small gap widths. The radii of the particles are identical. Potential theory is applied and the problem of interaction brought to the solution of an integrodifferential equation for radial velocity at the droplet/ thin layer interface. Asymptotic formulae are reported for interface velocity and drag force for small and large viscosity ratios. The results are compared with the solutions for a solid sphere and flat tangentially mobile interface. Thus, two major effects may be distinguished: opening of the gap and curving of the streamlines inside the droplets. The latter is shown to cause a relative increase in the drag force.     

Quantum Liouville Equation

The symmetrized product of quantum observables is defined. It is seen as consisting of ordinary multiplication followed by application of the superoperator that orders the operators of coordinate and momentum. This superoperator is defined in the way that allows obstruction free quantization of algebra of observables as well as introduction of operator version of the Poisson bracket. It is shown that this bracket has all properties of the Lie bracket and that it can substitute the commutator in the von Neumann equation leading to quantum Liouville equation.     

Hamiltonian Formulation of Statistical Ensembles and Mixed States of Quantum and Hybrid Systems

Representation of quantum states by statistical ensembles on the quantum phase space in the Hamiltonian form of quantum mechanics is analyzed. Various mathematical properties and some physical interpretations of the equivalence classes of ensembles representing a mixed quantum state in the Hamiltonian formulation are examined. In particular, non-uniqueness of the quantum phase space probability density associated with the quantum mixed state, Liouville dynamics of the probability densities and the possibility to represent the reduced states of bipartite systems by marginal distributions are discussed in detail. These considerations are used to study ensembles of hybrid quantum-classical systems. In particular, nonlinear evolution of a single hybrid system in a pure state and unequal evolutions of initially equivalent ensembles are discussed in the context of coupled hybrid systems.